Quadratic Equations: Solving by Factorising, Completing the Square and the Quadratic Formula | 二次方程：因式分解、配方法与求根公式解法

📚 Quadratic Equations: Solving by Factorising, Completing the Square and the Quadratic Formula | 二次方程：因式分解、配方法与求根公式解法

Quadratic equations appear throughout the IGCSE Maths syllabus, from algebra and graphs to area problems and projectile motion. Being able to solve them confidently by factorising, completing the square, or using the quadratic formula is a core skill that can earn marks quickly in both paper and non-paper assessments.

二次方程贯穿 IGCSE 数学大纲，从代数、图像到面积问题和抛体运动都有出现。能够熟练地通过因式分解、配方法或求根公式解二次方程是一项核心技能，在笔试和非笔试评估中都能快速得分。

1. What is a Quadratic Equation? | 什么是二次方程

ion a ≠ 0 is essential because if a = 0 the equation becomes linear rather than quadratic.

二次方程是次数为 2 的多项式方程。在单变量情况下，它可以写成标准形式 ax² + bx + c = 0，其中 a、b、c 是常数且 a ≠ 0。a ≠ 0 这个条件非常关键，因为如果 a = 0，方程就变成一次方程而不是二次方程。

equation x² + x = 6 can be rearranged into standard form by subtracting 6 from both sides.

例如，3x² − 5x + 2 = 0 和 x² − 9 = 0 都是二次方程。方程 x² + x = 6 可以通过两边减去 6 化为标准形式。

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2. Standard Form and Key Terms | 标准形式与关键术语

the quadratic term, bx is the linear term, and c is the constant term. The coefficients a, b and c can be positive, negative, zero or fractional, but a cannot be zero.

在 ax² + bx + c = 0 中，ax² 称为二次项，bx 称为一次项，c 称为常数项。系数 a、b、c 可以是正数、负数、零或分数，但 a 不能为零。

Before solving, always rearrange the equation so that one side equals 0. This makes factorising and applying the formula much easier.

在求解之前，一定要先把方程整理成一边等于 0 的形式。这样因式分解和使用公式都会容易得多。

Equation	a	b	c
2x² + 3x − 5 = 0	2	3	−5
x² − 4 = 0	1	0	−4
−x² + 2x = 0	−1	2	0

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3. Solving by Factorising | 因式分解法

roduct of two linear brackets. The equation ax² + bx + c = 0 becomes (px + q)(rx + s) = 0.

当二次表达式可以写成两个一次括号的乘积时，因式分解通常是最快的方法。方程 ax² + bx + c = 0 变成 (px + q)(rx + s) = 0。

7x + 12 = 0 factorises to (x + 3)(x + 4) = 0 because 3 and 4 multiply to 12 and add to 7.

例如，x² + 7x + 12 = 0 可以因式分解为 (x + 3)(x + 4) = 0，因为 3 和 4 相乘得 12，相加得 7。

uch as 2x² + 5x − 3 = 0, use decomposition or trial and error to find (2x − 1)(x + 3) = 0.

对于 a ≠ 1 的方程，如 2x² + 5x − 3 = 0，可以利用拆分法或试凑法得到 (2x − 1)(x + 3) = 0。

x² + 5x − 3 = (2x − 1)(x + 3)

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4. Zero Product Property | 零乘积性质

actor must be zero. Therefore, to solve (px + q)(rx + s) = 0, set each bracket equal to 0.

二次式因式分解后，我们使用零乘积性质：如果两个因式的乘积为零，那么至少有一个因式为零。因此，要解 (px + q)(rx + s) = 0，就令每个括号分别等于 0。

is gives x + 3 = 0 or x + 4 = 0, so x = −3 or x = −4. Always write both solutions clearly.

对于 (x + 3)(x + 4) = 0，令 x + 3 = 0 或 x + 4 = 0，所以 x = −3 或 x = −4。一定要把两个解都清楚地写出来。

peated solution is when the factor is a perfect square, such as (x − 2)² = 0 giving x = 2.

有些学生忘记二次方程通常有两个解。只有当因式为完全平方时，它才有一个重根解，例如 (x − 2)² = 0 得到 x = 2。

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5. Completing the Square | 配方法

Completing the square is a method that rewrites the quadratic expression in the form a(x + p)² + q. This is especially useful when factorising is difficult or when the question asks for the vertex of a parabola.

配方法是一种把二次表达式改写为 a(x + p)² + q 形式的方法。当因式分解困难或题目要求抛物线顶点时，尤其有用。

x to get 3, square it to get 9, then write x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4.

对于 x² + 6x + 5，把 x 的系数减半得 3，平方得 9，然后写成 x² + 6x + 5 = (x + 3)² − 9 + 5 = (x + 3)² − 4。

)² − 4 = 0, so (x + 3)² = 4. Taking square roots gives x + 3 = ±2, hence x = −1 or x = −5.

求解时，令 (x + 3)² − 4 = 0，所以 (x + 3)² = 4。开平方得 x + 3 = ±2，因此 x = −1 或 x = −5。

If the coefficient of x² is not 1, factor it out from the first two terms before completing the square.

如果 x² 的系数不是 1，先把它从前两项中提出来，再进行配方。

x² + 6x + 5 = (x + 3)² − 4

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6. The Quadratic Formula | 求根公式

The quadratic formula gives the solutions of ax² + bx + c = 0 as x = (−b ± √b² − 4ac) ÷ (2a). It works for any quadratic equation, including those that cannot be factorised neatly.

求根公式给出 ax² + bx + c = 0 的解为 x = (−b ± √b² − 4ac) ÷ (2a)。它适用于任何二次方程，包括那些不能很好因式分解的方程。

ntify a = 2, b = −4, c = −6. Substituting gives x = (4 ± √((−4)² − 4 × 2 × (−6))) ÷ (2 × 2

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